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Stochastic inflaton wave equation from an expanding environment

We discuss the inflaton $ϕ$ in an environment of scalar fields $χ_{n}$ on flat and curved manifolds. We average over the environmental fields $χ_{n}$. We study a contribution of superhorizon $k<<aH$ as well as subhorizon $k>> aH$ modes $χ_{n}({\bf k})$. As a result we obtain a stochastic wave equation with a friction and noise. We show that in the subhorizon regime in field theory a finite number of fields is sufficient to produce a friction and diffusion owing to the infinite number of degrees of freedom corresponding to different ${\bf k}$ in $χ_{n}({\bf k})$. We investigate the slow roll and the Markovian approximaions to the stochastic wave equation. A determination of the metric from the stochastic Einstein-Klein-Gordon equations is briefly discussed,